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# Roots

0
4
2
+1066

Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a+7}+\frac{1}{b+7}.$

Jan 14, 2024

#1
+289
+1

Simplify:

3x^2 - 11x - 7 = 0

Finding the roots by plugging in the values a = 3, b = -11, and c = -7 into the quadratic equation $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$, we get:

x = $$\frac { \sqrt{205} + 11}{6}$$, x = $$\frac { -\sqrt{205} + 11}{6}$$

We now have that a = $$\frac { \sqrt{205} + 11}{6}$$ and b = $$\frac { -\sqrt{205} + 11}{6}$$.

Plugging in these values for the expression $$\frac{1}{a+7}+\frac{1}{b+7}$$, we get:

$$\frac{1}{\frac { \sqrt{205} + 11}{6}+7}+\frac{1}{\frac { -\sqrt{205} + 11}{6}+7}$$

Simplifying this expression, we get:

$$\frac{1}{\frac { \sqrt{205} + 53}{6}}+\frac{1}{\frac { -\sqrt{205} + 53}{6}}$$

Removing the 6 and multiplying 1 by the reciprocal of $${\frac { \sqrt{205} + 53}{6}}$$, we get:

$$\frac{6}{ \sqrt{205} + 53}+\frac{6}{ -\sqrt{205} + 53}$$

We can plug each of these terms in the calculator to get

$$0.089 + 0.155$$

Adding these two values, we get:

$$0.244$$

Jan 14, 2024
#2
+128732
+1

$$\frac{1}{a+7}+\frac{1}{b+7}.$$

Simplify  as

3x^2 -11x -7  =  0

We have the form mx^2 + nx + c  = 0

m =3 , n = -11   c = -7

Product of  roots   = ab = c/m   =  -7/3

Sum of  roots =

a + b = -n/m  =   11/3

1/ (a + 7)  +  1 /(b + 7)   =

[ a + 7 + b + 7 ] / [ (a + 7) (b + 7) ]  =

[ a + b + 14 ] / [ ab + 7 (a + b) + 49 ]  =

[ 11/3 + 14 ]  / [ -7/3 + 7(11/3) + 49 ]  =

[ 53/3 ] / [ 217 /3 ]  =

53 / 217

Jan 14, 2024